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# Playing It Smart: Explaining the house edge in double-zero roulette

28 April 2009

Most big casinos have a "single-zero" ("European") roulette table or two, along with a cluster of "double-zero" ("American") games. Minimum wagers are usually higher on the single-zero version but edge is less by a factor of nearly two (an exception will be cited later). Further, single-zero games offer benefits like more pandering from dealers and bosses, finer perks, and often a table in a pit with snob appeal. So, naturally, enquiring minds want to know if it's worth betting more to get the better game.

Here's the scoop on the edge difference. A single-zero wheel has 37 positions, numbered 0 and 1 through 36. A double-zero wheel has 38 positions: 0, 00, and 1 through 36. A "straight-up" bet on one spot has one way to win and 36 to lose in a single-zero game; the odds against winning are accordingly 36-to-1. Analogously, you're fighting 37-to-1 odds at double-zero tables. But the payoff in either situation is 35-to-1. House advantage arises from the margin between the odds of winning and the payoff.

The distinction between 36-to-1 and 37-to-1 may seem small. But it morphs into a big difference in the average fraction of the money bet that the joints get to keep. To evaluate edge, multiply the probability of winning times the payoff per dollar bet, then subtract the probability of losing times the dollar to be lost. Single-zero edge works out to (1/37)x35 - (36/37)x1; this reduces to (35/37) - (36/37) which is -1/37 or -2.70 percent, an average loss of 2.70 cents per dollar. Double-zero edge is (1/38)x35 - (37/38)x1; this reduces to (35/38) - (37/38) which equals -2/38 or -5.26 percent, an average loss of 5.26 cents per dollar.

Probabilities and payoffs lead to the same edge on most bets for each table configuration. For example, \$1 on a four-number corner wins \$8 four ways and loses \$1 the other 33 or 34 in single- and double-zero games, respectively. In the first case, edge is (4/37)x8 - (33/37)x1, reducing to (32/37) - (33/37), again -1/37 or -2.70 percent. In the second case, it's (4/38)x8 - (34/38)x1, reducing to (32/38) - (34/38), -2/38 or -5.26 percent as before.

If you'd bet at or over the single-zero minimum anyway, go for lower edge. The dilemma arises when you want to risk less. Then you should ask what you forfeit by fraternizing with the elite.

A typical choice might be minimum bets of \$10 at double- and \$25 at single-zero tables. Pretend you'll bet the minimum in any event. An experienced, motivated dealer might get 50 spins in an hour. Your own gross hourly wager will therefore be \$500 in one instance and \$1,250 in the other. Your "expected" loss per hour will be \$26.30 and \$33.75, respectively. On this basis, you might decide it's worth \$7.45 an hour to hobnob with the high rollers.

Most solid citizens gamble with finite bankrolls, however. And they focus on prospects for single sessions, not statistical averages. Make believe your stake is \$400 and, to use an extreme illustration, you bet the \$10 or \$25 minimum straight up. Your chance of busting out with 40 losses in a row at \$10 in a double-zero game is about 34 percent. The chance of biting the dust with 16 successive misses at \$25 on a single-zero table is 64 percent.

Spreading bets across multiple numbers reduces the volatility of the game, improving your chance of longevity. But the divergence, predicated principally on bet size, will still be present.

There's something else. The exception mentioned earlier. Roulette buffs with the good fortune to be in Atlantic City will find that "outside" even-money bets at double-zero tables (Even, Odd, High, Low, Red, and Black) lose half rather than completely when a spin results in 0 or 00. This cuts the edge from 5.26 to 2.63 percent. Double-zero players betting wholly on the outside in these games consequently face less hammer than do their single-zero counterparts. Spread your bets, say \$10 on Red and \$1 on each of five black spots. The overall edge on the money at risk will exceed that on a single-zero table, but will be below the 5.26 percent you'd otherwise be battling. Which helps prove what the celebrated songster, Sumner A Ingmark, ceremoniously asserted:

Many a bet that seems propitious,
May not deliver what one wishes,
And's wisely regarded as suspicious.

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Best of Alan Krigman
Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.